1. Shall we take the train or the car?
(. . . route choice on coupled spatial networks)
Richard G. Morris1 and Marc Barthelemy2
1 University
2 Institut
of Warwick, Gibbet Hill Road, Coventry, U. K.
de Physique Théorique, CEA-DSM, Saclay, France.
December 5th, 2013

2. Aside: subjects I am interested in. . .
FHr L





0.010
Oblate
Prolate
0.005
20
40
60
80
r
-0.005
-0.010
15
10
5
0
0
5
10
15
1

m

=80
=160
=240
=320
=400
=480
=560
=640
0.5
0
-0.5

-1
0
50
100
150
200
z
250
300
350
400

3. Approaches to complex networks and coupled /
interdependent networks
1. Everyone knows Networks 101, but what about Networks 102 ?
2. Coupled networks are a trendy but ill-defined sub-class of
complex networks.
3. Hard (read: impossible) to characterise a generic type of
behaviour associated to such a broad class of systems i.e.:
3.1 percolation-like.
3.2 cascading sandpile-like.
4. Better to be ‘problem-led’: defining a model in order to answer
specific questions about well defined / motivated systems.

4. Routing in 2D: fast-but-sparse vs. slow-but-dense
1. Start with generic questions: how robust are current
transport/routing infrastructures?
1.1 Centralised power generation → distributed renewable
generation.
1.2 Bandwidth changes in packet routing (e.g., internet or other
ICT).
1.3 Technology changes in transport systems (e.g., rail/road).
2. Consider specific characterisation of such systems: namely,
when two different modes are available: ‘fast but sparse’
(long-range) networks and ‘slow but dense’ (short-range)
networks.

5. Time is of the essence
Need two (as yet unspecified) networks that are at the same time
different, but also connected. . . imagine they share some (not all)
nodes.
Consider a (static) route assignment problem, based on travel time.
Use weighted-shortest-paths: for a spatial graph G (V , E ) where
V = {xi } ∀ xi ∈ R2 , 1 ≤ i ≤ n, the weight of undirected edge
(xi , xj ) is given by:
wij = |xi − xj | /v ,
(1)
where v is the ‘speed’ of each network (i.e., weights ∝ time).

6. Topology: a planar subdivision
Take each network to be a Delaunay triangulation DT (V ):
. . . where V = {Xi } such that the Xi are i.i.d random variables in
R2 , distributed uniformly within a disk of radius r .

7. Connecting the two networks
. . . construct the two triangulations DT (1) and DT (2) such that
V (2) ⊆ V (1) (. . . where β = |V (2) |/|V (1) | ≤ 1):
13
19
20
13
21
13
19
20
20
23
21
23
25
22
25
26
16
8
22
26
16
8
14
14
24
15
9
24
15
17
10
0
12
17
10
0
12
17
6
24
4
9
6
4
1
11
1
4
11
1
5
5
7
5
7
2
18
2
18
3
30
3
30
28
29
3
27
28
29
27
29
DT (1)
DT (2)
1
1
V = V (1) ,
E = E (1) ∪ E (2) .
1
7

8. Sources and sinks
. . . sources and sinks are now characterised by an
’origin-destination’ matrix Tij (of size |V (1) |2 ).
13
13
19
19
20
20
21
21
23
25
23
22
26
16
0
12
17
14
9
8
6
4
10
0
12
17
24
15
14
24
15
9
1
6
4
11
5
25
22
26
16
8
10
1
11
7
5
2
2
18
18
3
30
28
29
3
27
30
28
29
27
x ∗:
Start with a (directed) star-graph centred on
E = {(xi , x ∗ )} ∀ xi = x ∗ and rewire using the following
algorithm:
for each e ∈ E :
1
1
with probability p:
replace (xi , x ∗ ) with (xi , yi ),
where yi is chosen uniformly at random from V = V (1) .
7

9. System characterisation: coupling
. . . the ’coupling’ is now a consequence of system parameters:
p,
α = v (1) /v (2) ,
β = n(2) /n(1) ,
where α, β ≤ 1.
Define:
λ=
Tij
i=j
coupled
σij
,
σij
(2)
where σij is the total number of weighted shortest paths between
coupled
xi and xj , and σij
is the number of weighted shortest paths
that use both networks.
[Note: T is normalised, i.e.,
ij
Tij = 1.]

10. System characterisation: efficiency & utility
Average route distance:
τ=
¯
Tij wij .
(3)
i=j
Gini coefficient of betweeness-centrality:
b (e) =
Tij
i=j
1
G= ¯ 2
2b|E |
σij (e)
σij
|b(p) − b(q)|.
p,q∈E
(4)
(5)

11. Results
. . . represents an average over the ensemble defined by values of
α, β and p.
Fix β and systematically vary α and p.
3.0
0.90
0.85
G
Τ
2.5
2.0
p
p
p
p
p
0.80
0.75
1.5
0.70
1.0
0.2
0.4
0.6
Λ
0.8
0.2 0.4 0.6 0.8 1.0
Λ
0.0
0.2
0.4
0.6
0.8

12. Results: Gini-coefficient at low p
Heatmap of normalised betweeness-centrality: 1
0
almost monocentric origin-destination matrix.
0.90
G
0.85
0.80
0.75
0.70
α = 0.9
α = 0.1
Gini-coefficient unchanged by increased coupling.
0.2 0.4 0.6 0.8 1.0
Λ

13. Results: Gini-coefficient at high p
Heatmap of normalised betweeness-centrality: 1
0
almost random origin-destination matrix.
0.90
G
0.85
0.80
0.75
0.70
α = 0.9
α = 0.1
Gini-coefficient increased by increased coupling.
0.2 0.4 0.6 0.8 1.0
Λ

14. Results: system utility
Define system utility as F = τ + µ G .
¯
. . . where λ∗ is defined such that F (λ∗ ) is minima.
p 0.0
p 0.8
Τ
11.0
10.5
Λ
ΜG
11.5
F
12.0
10.0
0.2 0.4 0.6 0.8 1.0
Λ
1.6
1.4
1.2
1.0
0.8
0.6
p
0.4
0.0 0.2 0.4 0.6 0.8 1.0
p

15. Wrap-up
Summary
Simple toy model—analysed by simulation—exhibiting
unexpected behaviour.
Two regimes emerge p > p ∗ and p ≤ p ∗ : Optimisation of the
system relies on the routing behaviour.
Outlook
Interacting or coupled networks play a prominent role in
modern life.
Understanding and classifying the behaviour of such systems
is important.
Familiarity with statistics and quantitative analysis is
important but ’off-the-shelf’ physics models are often
unhelpful.
R. G. Morris and M. Barthelemy, Phys. Rev. Lett. 109 (2012)